Question
A major DVD rental chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5000 of the 20000 households in the area are equipped with DVD players. It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have DVD players. State the test of hypothesis that is of interest to the rental chain.
H0: p = 0.32 versus H1: p > 0.32
H0: p = 0.25 versus H1: p > 0.25
H0: p = 5000 versus H1: p > 5000
H0: p = 5000 versus H1: p > 5000
H0: p = 0.32 versus H1: p > 0.32
H0: p = 0.25 versus H1: p > 0.25
H0: p = 5000 versus H1: p > 5000
H0: p = 5000 versus H1: p > 5000
Answers
GPT-4o mini
To determine the appropriate hypothesis test for the problem, we first need to outline what the rental chain is interested in. They want to know if more than 5000 out of 20,000 households have DVD players.
Given that they are polling 300 randomly selected households and found that 96 of them have DVD players, we can calculate the proportion of households with DVD players from the poll:
\[
\hat{p} = \frac{96}{300} = 0.32
\]
Now, we need to state the null and alternative hypotheses properly:
- The null hypothesis \( H_0 \): represents the scenario where the proportion of households that have DVD players is 0.25 (which corresponds to 5000 DVD players out of 20000 households). Thus:
\[
H_0: p = 0.25
\]
- The alternative hypothesis \( H_1 \): represents the scenario where the proportion of households that have DVD players is greater than 0.25. Thus:
\[
H_1: p > 0.25
\]
Based on the outline of the problem, the correct hypothesis test for the rental chain's interest is:
\[
H_0: p = 0.25 \quad \text{versus} \quad H_1: p > 0.25
\]
This choice tests whether the proportion of households with DVD players is significantly greater than 25%, which would imply that the rental chain's condition about having more than 5000 equipped households is met.
Given that they are polling 300 randomly selected households and found that 96 of them have DVD players, we can calculate the proportion of households with DVD players from the poll:
\[
\hat{p} = \frac{96}{300} = 0.32
\]
Now, we need to state the null and alternative hypotheses properly:
- The null hypothesis \( H_0 \): represents the scenario where the proportion of households that have DVD players is 0.25 (which corresponds to 5000 DVD players out of 20000 households). Thus:
\[
H_0: p = 0.25
\]
- The alternative hypothesis \( H_1 \): represents the scenario where the proportion of households that have DVD players is greater than 0.25. Thus:
\[
H_1: p > 0.25
\]
Based on the outline of the problem, the correct hypothesis test for the rental chain's interest is:
\[
H_0: p = 0.25 \quad \text{versus} \quad H_1: p > 0.25
\]
This choice tests whether the proportion of households with DVD players is significantly greater than 25%, which would imply that the rental chain's condition about having more than 5000 equipped households is met.